\contentsline {chapter}{\numberline {1}Introduction}{1}
\contentsline {section}{\numberline {1.1}Quantum Information}{1}
\contentsline {section}{\numberline {1.2}Concepts}{5}
\contentsline {subsection}{\numberline {1.2.1}The density matrix}{5}
\contentsline {subsection}{\numberline {1.2.2}Qubits}{7}
\contentsline {subsection}{\numberline {1.2.3}The Bloch/Poincar\'e sphere}{9}
\contentsline {subsection}{\numberline {1.2.4}Qubit transformations}{10}
\contentsline {subsection}{\numberline {1.2.5}Multi-qubit states}{11}
\contentsline {subsubsection}{Entanglement}{11}
\contentsline {subsection}{\numberline {1.2.6}The Bell/CHSH inequalities}{14}
\contentsline {subsection}{\numberline {1.2.7}Quantum measurement}{15}
\contentsline {subsection}{\numberline {1.2.8}Two-photon interference}{16}
\contentsline {subsection}{\numberline {1.2.9}Bell state filtering}{17}
\contentsline {section}{\numberline {1.3}Summary}{18}
\contentsline {chapter}{\numberline {2}Methods}{20}
\contentsline {section}{\numberline {2.1}Introduction}{20}
\contentsline {section}{\numberline {2.2}Quantum state tomography}{20}
\contentsline {subsection}{\numberline {2.2.1}Properties of the density matrix}{23}
\contentsline {subsection}{\numberline {2.2.2}Measuring the density matrix}{23}
\contentsline {subsubsection}{Linear inversion}{25}
\contentsline {subsubsection}{Error analysis}{28}
\contentsline {subsection}{\numberline {2.2.3}The positivity constraint}{30}
\contentsline {subsection}{\numberline {2.2.4}Maximum-likelihood estimation}{31}
\contentsline {subsubsection}{The likelihood function}{32}
\contentsline {subsection}{\numberline {2.2.5}An example: Two-photon polarization density matrix}{34}
\contentsline {section}{\numberline {2.3}Experimental state estimation}{36}
\contentsline {subsection}{\numberline {2.3.1}Spontaneous parametric downconversion}{37}
\contentsline {subsection}{\numberline {2.3.2}Polarization-entangled SPDC sources}{41}
\contentsline {subsection}{\numberline {2.3.3}Real-world sources}{42}
\contentsline {subsection}{\numberline {2.3.4}Type-I sandwich source}{43}
\contentsline {subsubsection}{Longitudinal walkoff}{43}
\contentsline {subsubsection}{Transverse walkoff}{45}
\contentsline {subsubsection}{Alignment of two crystals}{49}
\contentsline {subsection}{\numberline {2.3.5}Source output}{51}
\contentsline {subsection}{\numberline {2.3.6}Type-II beam-like emission source}{51}
\contentsline {section}{\numberline {2.4}Photon detection}{53}
\contentsline {subsection}{\numberline {2.4.1}Coincidence detection}{55}
\contentsline {section}{\numberline {2.5}Polarization manipulation}{57}
\contentsline {subsection}{\numberline {2.5.1}Polarization projection}{57}
\contentsline {section}{\numberline {2.6}Polarization rotators}{59}
\contentsline {subsection}{\numberline {2.6.1}Waveplates}{60}
\contentsline {subsection}{\numberline {2.6.2}Motorized rotation stages}{60}
\contentsline {subsection}{\numberline {2.6.3}Calibration}{62}
\contentsline {subsection}{\numberline {2.6.4}Liquid crystal waveplates}{65}
\contentsline {subsection}{\numberline {2.6.5}Liquid crystal waveplate driver}{67}
\contentsline {section}{\numberline {2.7}Polarization analysis}{68}
\contentsline {section}{\numberline {2.8}Two-photon interference}{71}
\contentsline {subsection}{\numberline {2.8.1}Alignment}{71}
\contentsline {subsubsection}{Iris scanning}{72}
\contentsline {subsection}{\numberline {2.8.2}Dips and antidips}{74}
\contentsline {subsection}{\numberline {2.8.3}Crystal walkoff}{75}
\contentsline {subsection}{\numberline {2.8.4}Type-II apparatus}{78}
\contentsline {section}{\numberline {2.9}Summary}{79}
\contentsline {chapter}{\numberline {3}Hidden differences}{80}
\contentsline {section}{\numberline {3.1}Two-mode systems}{82}
\contentsline {subsection}{\numberline {3.1.1}Characterizing two-mode states of light}{83}
\contentsline {subsection}{\numberline {3.1.2}Symmetric measurements}{86}
\contentsline {subsection}{\numberline {3.1.3}Group theory}{87}
\contentsline {subsubsection}{The symmetric group}{87}
\contentsline {subsubsection}{Representations}{88}
\contentsline {subsubsection}{Counting the irreps}{91}
\contentsline {subsubsection}{The Schur-Weyl duality}{93}
\contentsline {section}{\numberline {3.2}Hidden information}{100}
\contentsline {subsection}{\numberline {3.2.1}Structure of the accessible density matrix}{102}
\contentsline {subsection}{\numberline {3.2.2}Implications for distinguishability}{104}
\contentsline {section}{\numberline {3.3}Experimental measurements}{106}
\contentsline {section}{\numberline {3.4}The two-photon experimental polarization density matrix}{108}
\contentsline {subsection}{\numberline {3.4.1}Measuring the two-photon experimental density matrix}{111}
\contentsline {subsection}{\numberline {3.4.2}Measuring the three-photon accessible density matrix}{118}
\contentsline {section}{\numberline {3.5}Summary and conclusions}{122}
\contentsline {chapter}{\numberline {4}QST with MUBs}{124}
\contentsline {section}{\numberline {4.1}Introduction}{124}
\contentsline {section}{\numberline {4.2}Comparing different tomography strategies}{128}
\contentsline {section}{\numberline {4.3}Convergence of the estimate towards the true density matrix}{129}
\contentsline {section}{\numberline {4.4}The choice of measurement bases}{133}
\contentsline {section}{\numberline {4.5}Constructing MUBs for two qubits}{138}
\contentsline {section}{\numberline {4.6}Experiment}{139}
\contentsline {subsection}{\numberline {4.6.1}Separable measurements}{139}
\contentsline {subsection}{\numberline {4.6.2}MUBs measurements}{141}
\contentsline {subsection}{\numberline {4.6.3}Analysis}{144}
\contentsline {subsection}{\numberline {4.6.4}Results}{144}
\contentsline {section}{\numberline {4.7}The discrete Wigner function for two qubits}{148}
\contentsline {subsection}{\numberline {4.7.1}Finite fields}{149}
\contentsline {subsection}{\numberline {4.7.2}Discrete Wigner functions}{151}
\contentsline {subsection}{\numberline {4.7.3}Reconstruction}{152}
\contentsline {section}{\numberline {4.8}Summary}{153}
\contentsline {chapter}{\numberline {5}Measuring figures of merit}{155}
\contentsline {section}{\numberline {5.1}Introduction}{155}
\contentsline {section}{\numberline {5.2}Multiqubit measurements}{158}
\contentsline {section}{\numberline {5.3}Experimental techniques}{162}
\contentsline {subsection}{\numberline {5.3.1}Singlet state projection}{162}
\contentsline {subsection}{\numberline {5.3.2}Making impure states}{165}
\contentsline {section}{\numberline {5.4}Results}{168}
\contentsline {subsection}{\numberline {5.4.1}Density matrices}{168}
\contentsline {section}{\numberline {5.5}Discussion}{172}
\contentsline {chapter}{\numberline {6}Conclusions}{175}
\contentsline {section}{\numberline {6.1}Major results}{175}
\contentsline {section}{\numberline {6.2}Future outlook}{176}
\contentsline {chapter}{Bibliography}{178}
\contentsline {chapter}{\numberline {A}Electronic circuit designs}{193}
